A Banach space $X$ is called if for any cover $𝒰$ of $X$ by weakly open sets there is a finite subfamily $𝒱\subset 𝒰$ covering some ball of radius 1 centered at a point $x$ with $\parallel x\parallel \le r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty$-reflexive ($r$-reflexive for some $r\in \mathbb{N}$) if and only if each $\epsilon$-net for $X$ has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of $X$. We show that the quasireflexive James space $J$ is $r$-reflexive for no $r\in \mathbb{N}$. We do not know...

In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) $u$ for which $Ex(u,n)=O\left(n\right)$. The function $Ex(u,n)$ measures the maximum length of finite sequences over $n$ symbols which contain no subsequence of the type $u$. It follows from the result of Hart and Sharir that the containment $ababa\prec u$ is a (minimal) obstacle to $Ex(u,n)=O\left(n\right)$. We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth. In the second part of the paper we investigate...

Let $\Psi \left(\Sigma \right)$ be the Isbell-Mr’owka space associated to the $MAD$-family $\Sigma$. We show that if $G$ is a countable subgroup of the group $𝐒\left(\omega \right)$ of all permutations of $\omega$, then there is a $MAD$-family $\Sigma$ such that every $f\in G$ can be extended to an autohomeomorphism of $\Psi \left(\Sigma \right)$. For a $MAD$-family $\Sigma$, we set $Inv\left(\Sigma \right)=\{f\in 𝐒(\omega ):f[A]\in \Sigma$ for all $A\in \Sigma \}$. It is shown that for every $f\in 𝐒\left(\omega \right)$ there is a $MAD$-family $\Sigma$ such that $f\in Inv\left(\Sigma \right)$. As a consequence of this result we have that there is a $MAD$-family $\Sigma$ such that $n+A\in \Sigma$ whenever $A\in \Sigma$ and $n<\omega$, where $n+A=\{n+a:a\in A\}$ for $n<\omega$. We also notice that there is no $MAD$-family...